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Question: signals and systems signal processing...

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Signals and systems. Signal processing.

Question II In this problem you may use without proof the following formula for the sum of a geometric series with first term a, and common ratio r: N+1 ara You may also use the z-transform table provided (a) (50 points) (10 points) Define the discrete-time impulse function 6n) and prove that the z-transforn ii. (20 points) Prove that the z-transform of bnu n is and state and explain the region ii. (20 points) Consider a discrete time signal y/n], such that y/n-0,n <0. Prove that the of δ[n] is I. (Note that δ[n] as used here denotes the Kronecker delta function). of convergence. (Note that ufn] represents the discrete time unit step function) z-transform of the signal yln 1 is given by where Y(z) is the z-transform of yln] b (50 points) Consider the system with input n) and output y|n] given by y[n] = 0, n < 0; y[0] = 54, 546 where a and b are scalar constants (10 points) Is the system causal and time-invariant? Explain i. (5 points) Show that the transfer function of the system is giver Y(z)b X(z) -a G(z) =-=- ii. (15 points) Find the magnitude and phase of the frequency response of the filter associated with G(z), giving your answers in terms of the parameters a and b iv. (5 points) For what values of a is the system BIBO stable? Explain v. (15 points) Using a recursion method or a z-transform approach (or any other method), obtain a closed-form solution for y[n] when a-1.0075, b--1 and x[n] is the unit step u[n]

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